# The Wiener Index of Prime Graph $PG(\mathbb{Z}_n)$

## Authors

• Noor Hidayat Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Brawijaya
• Vira Hari Krisnawati Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Brawijaya
• Muhammad Husnul Khuluq Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Brawijaya
• Farah Maulidya Fatimah Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Brawijaya
• Ayunda Faizatul Musyarrofah Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Brawijaya

## Keywords:

Distance matrix, prime graph, ring, Wiener index

## Abstract

Let G = (V, E) be a simple graph and (R, +, ·) be a ring with zero element 0R. The Wiener index of G, denoted by W(G), is defined as the sum of distances of every vertex u and v, or half of the sum of all entries of its distance matrix. The prime graph of R, denoted by P G(R), is defined as a graph with V (P G(R)) = R such that uv ∈ E(P G(R)) if and only if uRv = {0R} or vRu = {0R}. In this article, we determine the Wiener index of P G(Zn) in some cases n by constructing its distance matrix. We partition the set Zn into three types of sets, namely zero sets, nontrivial zero divisor sets, and unit sets. There are two objectives to be achieved. Firstly,
we revise the Wiener index formula of P G(Zn) for n = p2 and n = p3 for prime number p and we compare this results with the results carried out by previous researchers. Secondly, we determine the Wiener index formula of P G(Zn) for n = pq, n = p2q, n =p 2q2, and n = pqr for distinct prime numbers p, q, and r.

## Author Biography

• Vira Hari Krisnawati, Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Brawijaya

Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Brawijaya

The Wiener Index of Prime Graph $PG(\mathbb{Z}_n)$. (2024). European Journal of Pure and Applied Mathematics, 17(3), 1659-1673. https://doi.org/10.29020/nybg.ejpam.v17i3.5166